Extensions of two dimensional lattice models of proteins have been developed to better understand their evolution and general biophysical properties. Previous highly idealized lattice models for proteins developed by Dill have been shown to display many of the general properties of proteins. Previous work by us showed that these models also display evolutionary behavior which parallels that of biological sequences. The current work generalizes these lattice models to larger alphabets (i.e., more different classes of residues), and subsequently more complex interaction rules. Evaluating this system using the previous criteria for folders indicates that such sequences have higher probabilities of folding to unique structures, and more different structures are obtained. This is in the context of effectively zero temperature. Moving to nonzero temperatures increases the realism of the model and an information theoretic framework is developed which evaluates the specificity of sequence for structure. In addition, a simple probabilistic model is developed which provides a basis for generating optimal interaction rules with respect to maximizing the information of the sequence-to-structure mapping. It is found that for realistic temperatures and low average interaction energies (in the range found in proteins), a single binary interaction corresponding to domination by the hydrophobic effect is optimal.